Matchings in metric spaces, the dual problem and calibrations modulo 2

TL;DR

This paper establishes a duality framework for matchings in metric spaces with an even number of points, introducing a 1-Lipschitz map to tree-like spaces, and extends these ideas to infinite spaces with a new notion of matching dimension.

Contribution

It introduces a duality approach for matchings in metric spaces, including a 1-Lipschitz mapping to trees and a concept of matching dimension, extending to infinite spaces.

Findings

Existence of a 1-Lipschitz map to a tree-like space preserving matching number

First version of unoriented Kantorovich duality for metric spaces

Extension of results to infinite metric spaces and introduction of matching dimension

Abstract

We show that for a metric space with an even number of points there is a 1-Lipschitz map to a tree-like space with the same matching number. This result gives the first basic version of an unoriented Kantorovich duality. The study of the duality gives a version of global calibrations for 1-chains with coefficients in $\mathbb Z_2$. Finally we extend the results to infinite metric spaces and present a notion of "matching dimension" which arises naturally.